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A Refined scissors congruence group and the third homology of $\textrm{SL}_2$

Behrooz Mirzaii, Elvis Torres Pérez

Abstract

There is a natural connection between the third homology of $\textrm{SL}_2(A)$ and the refined Bloch group $\mathcal{RB}(A)$ of a commutative ring $A$. In this article we investigate this connection and as the main result we show that if $A$ is a universal $\textrm{GE}_2$-domain such that $-1 \in A^{\times 2}$, then we have the exact sequence $H_3(\textrm{SM}_2(A),\mathbb{Z}) \to H_3(\textrm{SL}_2(A),\mathbb{Z}) \to \mathcal{RB}(A) \to 0$, where $\textrm{SM}_2(A)$ is the group of monomial matrices in $\textrm{SL}_2(A)$. Moreover we show that $\mathcal{RP}_1(A)$, the refined scissors congruence group of $A$, naturally is isomorph with the relative homology group $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$.

A Refined scissors congruence group and the third homology of $\textrm{SL}_2$

Abstract

There is a natural connection between the third homology of and the refined Bloch group of a commutative ring . In this article we investigate this connection and as the main result we show that if is a universal -domain such that , then we have the exact sequence , where is the group of monomial matrices in . Moreover we show that , the refined scissors congruence group of , naturally is isomorph with the relative homology group .
Paper Structure (8 sections, 20 theorems, 184 equations)

This paper contains 8 sections, 20 theorems, 184 equations.

Table of Contents

  1. The Lg-rings and the complex of unimodular vectors
  2. The refined scissors congruence group
  3. The map Lg
  4. The refined Bloch group
  5. The low dimensional homology of Lg
  6. The third homology of Lg
  7. A spectral sequence for relative homology
  8. The groups Lg and Lg

Key Result

Proposition 1.1

Let $A$ be a commutative ring. (i) The ring $A$ satisfies the condition that $X_\bullet(A^2) \rightarrow \mathbb{Z}$ is exact in dimension $<1$ if and only if $A$ is a ${\rm GE}_2$-ring. (ii) If $A$ is universal for ${\rm GE}_2$, then $X_\bullet(A^2)$ is exact in dimension $1$, i.e. $H_1(X_\bullet(

Theorems & Definitions (50)

  • Proposition 1.1: Hutchinson
  • proof
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Example 3.3
  • ...and 40 more